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0580/41 May/Jun 2024, Question 7: Volume and Surface Area of a Cone

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 15 marks

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The question, in short

Use a 70 cm by 30 cm by 40 cm open-top cuboid box to find its internal surface area and how many 20 cm by 15 cm cylinders fit inside; then use a 750 g bronze cone with density 8.9 g/cm³ and radius:height 1:3 to show r = 2.99 cm and find its total surface area.

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Part (ai) · 3 marks

Calculate the total area of the inside surfaces of the open box, excluding the top.

  1. 1Base = 70 × 30 = 2100
  2. 2Two ends = 2 × 30 × 40 = 2400
  3. 3Two sides = 2 × 40 × 70 = 5600
  4. 4Total = 2100 + 2400 + 5600 = 10 100 cm²

Answer10 100

Part (aii) · 3 marks

Determine the greatest number of cylinders, each 20 cm high with diameter 15 cm, that fit completely in the box.

  1. 1Length: 70 ÷ 15 = 4.67 → 4 cylinders
  2. 2Width: 30 ÷ 15 = 2
  3. 3Height: 40 ÷ 20 = 2
  4. 4Maximum = 4 × 2 × 2 = 16

Answer16

Part (bi) · 4 marks

Using mass 750 g, density 8.9 g/cm³ and radius:height 1:3, show the cone radius is 2.99 cm to 3 significant figures.

  1. 1Volume = 750 ÷ 8.9 = 84.27 cm³
  2. 2h = 3r, so 1/3 × π × r² × 3r = πr³
  3. 3πr³ = 84.27
  4. 4r³ = 84.27 ÷ π = 26.82
  5. 5r = ∛26.82 = 2.993… = 2.99 cm (3 s.f.)

Answerr = 2.993… = 2.99 cm

Part (bii) · 5 marks

Find the cone's total surface area, including the base, using its radius and slant height.

  1. 1h = 3 × 2.99 = 8.97
  2. 2l = √(2.99² + 8.97²) = √89.40 = 9.455
  3. 3Base = π × 2.99² = 28.09
  4. 4Curved = π × 2.99 × 9.455 = 88.82
  5. 5Total = 28.09 + 88.82 = 116.9 ≈ 117 cm²

Answer117 or 116.9 to 117.2

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Also searched as: 0580/41/M/J/24 · 0580_s24_qp_41 · 0580 s24 41 · question 7.
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